Problem: Factor the following expression: $-5$ $x^2+$ $4$ $x+$ $12$
Answer: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-5)}{(12)} &=& -60 \\ {a} + {b} &=& & & {4} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-60$ and add them together. Remember, since $-60$ is negative, one of the factors must be negative. The factors that add up to ${4}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-6}$ and ${b}$ is ${10}$ $ \begin{eqnarray} {ab} &=& ({-6})({10}) &=& -60 \\ {a} + {b} &=& {-6} + {10} &=& 4 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-5}x^2 {-6}x +{10}x +{12} $ Group the terms so that there is a common factor in each group: $ ({-5}x^2 {-6}x) + ({10}x +{12}) $ Factor out the common factors: $ x(-5x - 6) - 2(-5x - 6) $ Notice how $(-5x - 6)$ has become a common factor. Factor this out to find the answer. $(-5x - 6)(x - 2)$